examples.tex

   1 \documentclass{report}
   2
   3 % Setting hypertexnames=false forces hyperref to use a consistent
   4 % internal counter for proposition/equation references rather than
   5 % being clever, which doesn't work after we reset those counters.
   6 \usepackage[hypertexnames=false]{hyperref}
   7 \hypersetup{
   8   colorlinks=true,
   9   linkcolor=blue,
  10   citecolor=blue
  11 }
  12
  13 % We have to load this after hyperref, so that links work, but before
  14 % mjotex so that mjotex knows to define its glossary entries.
  15 \usepackage[nonumberlist]{glossaries}
  16 \makenoidxglossaries
  17
  18 % If you want an index, we can do that too. You'll need to define
  19 % the "INDICES" variable in the GNUmakefile, though.
  20 \usepackage{makeidx}
  21 \makeindex
  22
  23 \usepackage{mjotex}
  24 \usepackage{mathtools}
  25
  26 \begin{document}
  27
  28   \begin{section}{Algebra}
  29     If $R$ is a \index{commutative ring}, then $\polyring{R}{X,Y,Z}$
  30     is a multivariate polynomial ring with indeterminates $X$, $Y$,
  31     and $Z$, and coefficients in $R$. If $R$ is a moreover an integral
  32     domain, then its fraction field is $\Frac{R}$.
  33   \end{section}
  34
  35   \begin{section}{Algorithm}
  36     An example of an algorithm (bogosort) environment.
  37
  38     \begin{algorithm}[H]
  39       \caption{Sort a list of numbers}
  40       \begin{algorithmic}
  41         \Require{A list of numbers $L$}
  42         \Ensure{A new, sorted copy $M$ of the list $L$}
  43
  44         \State{$M \gets L$}
  45
  46         \While{$M$ is not sorted}
  47           \State{Rearrange $M$ randomly}
  48         \EndWhile
  49
  50         \Return{$M$}
  51       \end{algorithmic}
  52     \end{algorithm}
  53   \end{section}
  54
  55   \begin{section}{Arrow}
  56     The identity operator on $V$ is $\identity{V}$. The composition of
  57     $f$ and $g$ is $\compose{f}{g}$. The inverse of $f$ is
  58     $\inverse{f}$. If $f$ is a function and $A$ is a subset of its
  59     domain, then the preimage under $f$ of $A$ is $\preimage{f}{A}$.
  60   \end{section}
  61
  62   \begin{section}{Calculus}
  63     The gradient of $f : \Rn \rightarrow \Rn[1]$ is $\gradient{f} :
  64     \Rn \rightarrow \Rn$.
  65   \end{section}
  66
  67   \begin{section}{Common}
  68     The function $f$ applied to $x$ is $f\of{x}$. We can group terms
  69     like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
  70     set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. Here's a pair
  71     of things $\pair{1}{2}$ or a triple of them $\triple{1}{2}{3}$,
  72     and the factorial of the number $10$ is $\factorial{10}$.
  73
  74     The Cartesian product of two sets $A$ and $B$ is
  75     $\cartprod{A}{B}$; if we take the product with $C$ as well, then
  76     we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$ and $W$
  77     is $\directsum{V}{W}$. Or three things,
  78     $\directsumthree{U}{V}{W}$. How about more things? Like
  79     $\directsummany{k=1}{\infty}{V_{k}} \ne
  80     \cartprodmany{k=1}{\infty}{V_{k}}$. Those direct sums and
  81     cartesian products adapt nicely to display equations:
  82     %
  83     \begin{equation*}
  84       \directsummany{k=1}{\infty}{V_{k}} \ne \cartprodmany{k=1}{\infty}{V_{k}}.
  85     \end{equation*}
  86     Here are a few common tuple spaces that should not have a
  87     superscript when that superscript would be one: $\Nn[1]$,
  88     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
  89     superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
  90     $\Qn[2]$, $\Rn[2]$, $\Cn[2]$.
  91
  92     We also have a few basic set operations, for example the union of
  93     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
  94     course with union comes intersection: $\intersect{A}{B}$,
  95     $\intersectthree{A}{B}{C}$. We can also take an arbitrary
  96     (indexed) union and intersections of things, like
  97     $\unionmany{k=1}{\infty}{A_{k}}$ or
  98     $\intersectmany{k=1}{\infty}{B_{k}}$. The best part about those
  99     is that they do the right thing in a display equation:
 100     %
 101     \begin{equation*}
 102       \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
 103     \end{equation*}
 104
 105     Finally, we have the four standard types of intervals in $\Rn[1]$,
 106     %
 107     \begin{align*}
 108       \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
 109       \intervaloc{a}{b} &= \setc{ x \in \Rn[1]}{ a < x \le b },\\
 110       \intervalco{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x < b }, \text{ and }\\
 111       \intervalcc{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x \le b }.
 112     \end{align*}
 113   \end{section}
 114
 115   \begin{section}{Complex}
 116     We sometimes want to conjugate complex numbers like
 117     $\compconj{a+bi} = a - bi$.
 118   \end{section}
 119
 120   \begin{section}{Cone}
 121     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
 122     are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.  If cones
 123     $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
 124     $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
 125     $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
 126     \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
 127     \ltcone_{K} y$ with respect to a cone $K$.
 128   \end{section}
 129
 130   \begin{section}{Convex}
 131     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
 132     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
 133     then its lineality space is $\linspace{K}$, its lineality is
 134     $\lin{K}$, and its extreme directions are $\Ext{K}$. The fact that
 135     $F$ is a face of $K$ is denoted by $F \faceof K$; if $F$ is a
 136     proper face, then we write $F \properfaceof K$.
 137   \end{section}
 138
 139   \begin{section}{Euclidean Jordan algebras}
 140     The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
 141     is $\jp{x}{y}$.
 142   \end{section}
 143
 144   \begin{section}{Font}
 145     We can write things like Carathéodory and Güler and $\mathbb{R}$.
 146   \end{section}
 147
 148   \begin{section}{Linear algebra}
 149     The absolute value of $x$ is $\abs{x}$, or its norm is
 150     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
 151     their tensor product is $\tp{x}{y}$. The Kronecker product of
 152     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
 153     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
 154     $\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific
 155     concept is the Moore-Penrose pseudoinverse of $L$, denoted by
 156     $\pseudoinverse{L}$.
 157
 158     The span of a set $X$ is $\spanof{X}$, and its codimension is
 159     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
 160     automorphism group of $X$ is $\Aut{X}$, and its Lie algebra is
 161     $\Lie{X}$. We can write a column vector $x \coloneqq
 162     \colvec{x_{1},x_{2},x_{3},x_{4}}$ and turn it into a $2 \times 2$
 163     matrix with $\matricize{x}$. To recover the vector, we use
 164     $\vectorize{\matricize{x}}$.
 165
 166     The set of all bounded linear operators from $V$ to $W$ is
 167     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
 168     instead.
 169
 170     If you want to solve a system of equations, try Cramer's
 171     rule~\cite{ehrenborg}.
 172
 173     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
 174     but what if $W = V^{\perp}$? Then we wish to indicate that fact by
 175     writing $\directsumperp{V}{W}$. That operator should survive a
 176     display equation, too, and the weight of the circle should match
 177     that of the usual direct sum operator.
 178     %
 179     \begin{align*}
 180       Z = \directsumperp{V}{W}\\
 181       \oplus \oplusperp \oplus \oplusperp
 182     \end{align*}
 183     %
 184     Its form should also survive in different font sizes...
 185     \Large
 186     \begin{align*}
 187       Z = \directsumperp{V}{W}\\
 188       \oplus \oplusperp \oplus \oplusperp
 189     \end{align*}
 190     \Huge
 191     \begin{align*}
 192       Z = \directsumperp{V}{W}\\
 193       \oplus \oplusperp \oplus \oplusperp
 194     \end{align*}
 195     \normalsize
 196   \end{section}
 197
 198   \begin{section}{Listing}
 199     Here's an interactive SageMath prompt:
 200
 201     \begin{tcblisting}{listing only,
 202                        colback=codebg,
 203                        coltext=codefg,
 204                        listing options={language=sage,style=sage}}
 205     sage: K = Cone([ (1,0), (0,1) ])
 206     sage: K.positive_operator_gens()
 207     [
 208     [1 0]  [0 1]  [0 0]  [0 0]
 209     [0 0], [0 0], [1 0], [0 1]
 210     ]
 211     \end{tcblisting}
 212
 213     However, the smart way to display a SageMath listing is to load it
 214     from an external file (under the ``listings'' subdirectory):
 215
 216     \sagelisting{example}
 217
 218     Keeping the listings in separate files makes it easy for the build
 219     system to test them.
 220   \end{section}
 221
 222   \begin{section}{Miscellaneous}
 223     The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
 224     = 3$.
 225   \end{section}
 226
 227   \begin{section}{Proof by cases}
 228
 229     \begin{proposition}
 230       There are two cases in the following proof.
 231
 232       \begin{proof}
 233         The result should be self-evident once we have considered the
 234         following two cases.
 235         \begin{pcases}
 236           \begin{case}[first case]
 237             Nothing happens in the first case.
 238           \end{case}
 239           \begin{case}[second case]
 240             The same thing happens in the second case.
 241           \end{case}
 242         \end{pcases}
 243
 244         You see?
 245       \end{proof}
 246     \end{proposition}
 247
 248     Here's another one.
 249
 250     \renewcommand{\baselinestretch}{2}
 251     \begin{proposition}
 252       Cases should display intelligently even when the document is
 253       double-spaced.
 254
 255       \begin{proof}
 256         Here we go again.
 257
 258         \begin{pcases}
 259           \begin{case}[first case]
 260             Nothing happens in the first case.
 261           \end{case}
 262           \begin{case}[second case]
 263             The same thing happens in the second case.
 264           \end{case}
 265         \end{pcases}
 266
 267         Now it's over.
 268       \end{proof}
 269     \end{proposition}
 270     \renewcommand{\baselinestretch}{1}
 271   \end{section}
 272
 273   \begin{section}{Theorems}
 274     \begin{corollary}
 275       The
 276     \end{corollary}
 277
 278     \begin{lemma}
 279       quick
 280     \end{lemma}
 281
 282     \begin{proposition}
 283       brown
 284     \end{proposition}
 285
 286     \begin{theorem}
 287       fox
 288     \end{theorem}
 289
 290     \begin{exercise}
 291       jumps
 292     \end{exercise}
 293
 294     \begin{definition}
 295       quod
 296     \end{definition}
 297
 298     \begin{example}
 299       erat
 300     \end{example}
 301
 302     \begin{remark}
 303       demonstradum.
 304     \end{remark}
 305   \end{section}
 306
 307   \begin{section}{Theorems (starred)}
 308     \begin{corollary*}
 309       The
 310     \end{corollary*}
 311
 312     \begin{lemma*}
 313       quick
 314     \end{lemma*}
 315
 316     \begin{proposition*}
 317       brown
 318     \end{proposition*}
 319
 320     \begin{theorem*}
 321       fox
 322     \end{theorem*}
 323
 324     \begin{exercise*}
 325       jumps
 326     \end{exercise*}
 327
 328     \begin{definition*}
 329       quod
 330     \end{definition*}
 331
 332     \begin{example*}
 333       erat
 334     \end{example*}
 335
 336     \begin{remark*}
 337       demonstradum.
 338     \end{remark*}
 339   \end{section}
 340
 341   \begin{section}{Topology}
 342     The interior of a set $X$ is $\interior{X}$. Its closure is
 343     $\closure{X}$ and its boundary is $\boundary{X}$.
 344   \end{section}
 345
 346   \setlength{\glslistdottedwidth}{.3\linewidth}
 347   \setglossarystyle{listdotted}
 348   \glsaddall
 349   \printnoidxglossaries
 350
 351   \bibliographystyle{mjo}
 352   \bibliography{local-references}
 353
 354   \printindex
 355 \end{document}